19 443 43 20 5. Required the product of 561 × 7 × X X 47 683 Ans. Logarithm of the product is 1.34735, and the product 22.251. 6. Required the product of 0594405 x 583′ x *0322916 × 428571 x 'l'8'. Ans. Log. of the product is -5.94074, and the product=000087244. 7. Divide 0565 by 25. Quotient 226. 8. Divide 00375 by 0678. Quotient 05531. 9. Divide 54498 by '093. Quotient = 586000. 10. Divide by 1538. Quotient = '0041962. 11. Involve 1.05 to the 40th power. Ans. 7·0405. 12. Required the 3.75 power of 14.79; or find the value 15 of 14.791 Ans. 24400. 13. Required the 34'54' power of 94.75'; or find the 14. Involve 09475' to the 34'54' power. Ans. ·44307. 15. Find the cube root of 000381078. Ans. 0725. 16. What is the 625 root of 027588? Ans. 0035606. 17. Find a fourth proportional to 58m 13; 11"75; and 24 hours. Ans. 4m 50.6. 18. Find a fourth proportional to 23h 12m 375; 24 hours; and 7h 59m 34s. Ans. 8h 15m 53s. CHAP. III. THE USE OF THE TABLES OF SINES AND TANGENTS.* PROPOSITION 1. (17) To find the natural sine or cosine of an arc, also the lorgarithmic sine, tangent, secant, &c. RULE. If the degrees in the arc be less than 45, look for them at the top of the table, and for the minutes (if any) in the left hand column marked ('), against which, in the column signed at the top of the table with the proposed name, viz. sine, cosine, &c. stands the sine, cosine, &c. required. If the degrees are more than 45, they must be found at the bottom of the table, and the minutes (if any) must be found in the right hand column. The name in this case, viz. sine, tangent, &c. must be taken at the bottom of the table. To find the secants, see the first page of Table III. *The construction of these tables will be found at the end of Book II. Chap. V. Before the student reads this and the following chapter, it will be proper for him to read the definitions, &c. in Book II. Chap. I. The natural sines must be looked for in the table entitled natural sines; and the logarithmic sines in the table entitled logarithmic sines and tangents. Required the natural and logarithm, sine and cosine of 39° 42'. Natural sine of 39° 42′ =63877, cosine=⚫76940. Logarithmic sine of 39° 42′ = 9·80534, cosine = 9.88615. Required the natural and logarithm. sine and cosine of 73° 27'. Natural sine of 73° 27′ = ·95857, cosine = '28485. Logarithmic sine of 73° 27′ = 9.98162, cosine = 9.45462. If the sine, tangent, &c. be wanted to any number of degrees above 90, subtract those degrees from 180° and find the sine, tangent, &c. of the remainder; or subtract 90° from the given number of degrees, and find the cosine, cotangent, &c. of the remainder, which is the same thing. Required the logarithmic sine, tangent, secant, cosine, cotangent, and cosecant of 137° 29′. 180° 137° 29' rem. 42° 31', sine 9.82982, cosine 9.86752, tangent = 9.96231, cotangent 10·03769, secant = 10·13248, cosecant = 10.17018, and these are respectively equal to the cosine, sine, cotangent, tangent, cosecant, and secant of 47° 29′ = 137° 29′. 90°. PROPOSITION II. (18) To find the logarith. sine, cosine, &c. of an arc to seconds. Find the logarithm to the degrees and minutes as in Proposition I.; take the difference between this logarithm and the next greater or less in the same column, according as you want a sine or cosine, tangent or cotangent, &c.; multiply this difference by the number of seconds given, and divide the product by 60; add the quotient to the given logarithm if it be a sine, tangent, or secant, but subtract the quotient from the given logarithm if it be a cosine, cotangent, or cosecant, and the sum, or remainder, will be the logarithm required. Required the logarith. sine, tangent, and secant of 35° 44′ 24′′. Log. sine 35° 44′9.76642 tan = 9.85700 sec = 10·09058 next greater sine 9.76660 tan = 9.85727 sec = 10.09067 = Then, sine 35° 44′ 24′′ 9.76649, tan = 9.85711, sec = 10.09062. In the same manner the natural sine is found, being ⚫58411. Required the logarithmic cosine, cotangent, and cosecant of 35° 44' 24". Log cos 35° 44'=9.90942 cotan=10·14300 cosec=10.23358 next less cosine=2.90933 cotan10·14273 cosec=10.23340 Then cosine 35° 44' 24" 9.90938, cotan 10·14289, cosecant 10.23351. In a similar manner the natural cosine is found, being ⚫81167. PROPOSITION III. (19) To find the degrees, minutes, or degrees, minutes, and seconds, corresponding to any given logarithmic sine, tangent, &c. RULE. Find the nearest logarithm to the given one in the table, and the degrees answering to it will be found at the top of the column if the name be there, and the minutes on the left hand; but if the name be at the bottom of the table, the degrees must be found at the bottom of the table, and the minutes on the right hand. To find the arc to seconds, take the difference between the two logarithms which include the given one, also the difference between the given logarithm and the next less. Multiply the latter difference by 60, and divide the product by the former difference, the quotient will give the number of seconds, which must be added to the degrees and minutes corresponding to the next less number in the tables, if your given logarithm be a sine, tangent, or secant; but if your given logarithm be a cosine, cotangent, or cosecant, the number of seconds must be subtracted from the degrees and minutes corresponding to the next less number in the tables. Find the degrees, minutes, and seconds, corresponding to the logarithmic sine 9.43299. Sine less than the given one 9.43278 21 60 Therefore the required arc is 15° 43′ 28′′. The same manner of proceeding must be observed in finding a tangent, secant, or natural sine. Find the degrees, minutes, and seconds, corresponding to the logarithmic cosine 9.43297. Cosine less than the given one 9.43278 19 Therefore the required arc is 74° 16′ 35'. PROPOSITION IV. (20) To find the natural or logarithmic versed sine of an arc, by the help of a table of natural or logarithmic sines. To find the natural versed sine, subtract the natural cosine from unity if the arc be less than 90°, but if greater than 90 add it to unity. To find the logarithmic versed sine, find the logarithmic sine of half an arc, double it, and subtract 9.69897 from the product. Required the natural versed sine Required the natural versed sine of 65° 45'. Radius=1 Nat cosine 65° 45′ 41072 Nat ver sine 65° 45′ = ·58928 of 115° 35'. Natural cosine 115° 35′ or cosine 64° 25′ = ·43182 To which add 1· vers sine of 115° 35'-1·43182 Required the logarithm. versed Required the log. versed sine of Log vers sine 72° 14' 9.84189 Log vers sine 37° 53′ =9.32374 CHAP. IV. THE CONSTRUCTION AND USE OF THE PLANE SCALE. (21) The Plane Scale is a mathematical instrument of extensive use. The scale generally used at sea is two feet in length, having drawn upon it equal parts, chords, sines, tangents, secants, &c. These are contained on one side of the scale, and the other side contains their logarithms. (22) Describe a semicircle with any convenient radius CB (Fig. I. Plate II.); from the centre c draw CD perpendicular to AB, and produce it to F, &c.; draw BE parallel to CF, and join AD and BD. (23) Rhumbs. Divide the quadrantal arc AD into eight equal parts, with one foot of the compasses in a, transfer the distances A1, A2, A3, &c. to the straight line AD, and it will be a line of rhumbs containing eight points of the compass, or one fourth of the whole circumference of the compass. By subdividing each of the divisions al; 1-2; &c. into four equal parts, and transferring them in the same manner to the line AD, it will contain the points, and half and quarter points. (24) Chords. Divide the arc BD into nine equal parts, with one foot of the compasses in B and the distances B-10, B-20, B-30, &c., transfer them to the straight line BD, which will be a line of chords constructed to every ten degrees. The single degrees are constructed by subdividing the arcs B-10, 10-20, &c. into ten equal parts, and transferring the divisions in the same manner to the line BD. (25) Sines. Through each of the divisions of the arc BD draw lines parallel to CD, such as 80-10, 70-20, &c., and the line CB will be divided into a line of sines reckoning from c to B (for CG is the cosine of the arc B-80, or the sine of the arc D-80, which is ten degrees); if this line be numbered from B towards c, it will become a line of versed sines. (26) Tangent. From the centre c draw straight lines through the several divisions of the quadrantal arc BD, to touch the straight line BE, which will become a line of tangents. (27) Secants. Transfer the distances between the centre c and the divisions of the line of tangents, to the line DF, and it will become a line of secants which must be numbered from D towards F, as in the figure. (28) Semi-tangents. From a draw lines through the several divisions of the arc BD, and they will divide the line CD into |